2.3
2.2
The Limit of a Function
Definition (Limit, informal version). For a function f and numbers a and L, we say that the limit of
f (x) as x approaches a is L if we can force the value of f (x) to be as close to L as we wish by considering
only values of x sufficiently close to a, but not equal to a. This is written as
lim f (x) = L
x!a
Note that the value of f (a) is irrelevant: f need not even be defined for x = a.
Examples 1-5,
pages 89-92
Exploring All Nearby x Values
Examples 2 and 4 of the text show that it is important to consider all values of x near a when
studying a limit as x ! a, not just a selection.
2x2 2
Example A. Show that lim f (x) = lim
= 4.
x!1
x!1 x
1
We can simplify to f (x) = 2x + 2, valid for all x 6= 1.
Then we measure how close two numbers are
by the absolute value of their difference. For example, if x is within 0.001 of 1, |x 1| < 0.001, and so
|f (x) 4| = |(2x + 2) 4| = |2x 2| = 2|x 1| which is less than 0.002. When we look only at x values
ever closer to 1, in that |x 1| is ever smaller, |f (x) 4 = 2|x 1| is ever smaller: f (x) gets ever closer
to 4. For example, the value f (x) is sure to be within a tiny 10 100 of 4 when we look at x values within
0.5 ⇥ 10 100 of 1. So the limit is 4.
Example B: A Function With a Jump Consider the function f (x) given by
8
2
>
2
< 2x
, x 6= 1
x 1
f (x) =
>
:
1,
x=1
What is the limit of f (x) as x approaches 1?
Since the limit as x ! a is based on values of f (x) for all x near a, but not equal to a, only the formula
(2x2 2)/(x 1) matters! And since it is equal to 2x+2 for all x values near 1, the value is near 2·1+2 = 4
there, and the limit is 4: limx!1 f (x) = 4, not 1.
Note well: The limit of f (x) as x goes to a does not always equal the value f (a), even when f (a)
makes sense!
The graph of this function has a jump at x = 1, but the limit calculation ignores this, and treats the function
as as if it were “uninterrupted” or “continuous” there.
Example 6 p. 92: Another type of jump: the Heaviside function
In the physical description of sudden changes, like turning on a power switch, the Heaviside Function is
often useful:
H(t) = 0 for t < 0, H(t) = 1 for t 0
For t near 0 and positive, H(t) is 1, suggesting a limit of 1.
But for t near 0 and negative, H(t) is 0, suggesting a limit of 0.
The limit cannot be both zero and one, so again this function has no limit as t ! 0, due to this jump from
one value to another, which breaks the graph at this point.
One-sided Limits
In the example above, we see that H(t) has “no limit” as t ! 0, but it is useful also to describe what
happens at times just before t = 0, and what happens at times just after t = 0: what happens to
one side or the other of a point on the graph.
T HE L IMIT OF A F UNCTION
2.4
We want to note that “as t approaches 0 from the right (t > 0), H(t) approaches 1.” We use
the notation t ! 0+ , with a plus sign superscript indicating that only t values to the right are
considered: the relevant t values are “0 plus something”. The value approached is the right-hand
limit, or the limit from the right, with short-hand notation
lim H(t) = 1.
t!0+
Similarly the behavior for t near 0 and less than zero is called the left-hand limit and we use a
minus sign superscript, because the t value is “0 minus something:”
lim H(t) = 0.
t!0
Note well: “t ! 0 ” is different from “t ! 0”, which would be a funny way of writing a normal
“two-sided” limit. And t ! 1 is very different than t ! 1; the former is about what happens for
t just below 1; the latter is about what happens for t near 1.
The limit exists when both one sided limits exist and agree
Comparing definitions, the limit of a function as x ! a exists exactly when both one sided limits
exist, and both give the same value. Sometimes, computing a limit one side at a time is easiest: in
particular when the function is given by different formulas on the two sides.
Example 7
Infinite Limits
We have seen several ways that a function can fail to have a limit as x ! a, and decided that
sometimes, there is still something useful to say about how the function behaves for x near a (onesided limits). Example 8, page 94 gives another case of that: trying to compute the limit of 1/x2 as
x ! 0.
This is not a normal limit with a real number as its value, as discussed above: we say that 1/x2 has
an infinite limit at x = 0.
Of course the values of f (x) could also go the opposite way, down to ever lower values with no
1
lower bound. For example, we say that lim
= 1.
x!1 (x
1)2
One-sided Infinite Limits
Finally, it is natural to combine the ideas of infinite limits and one-sided limits.
1
Exercise C. Describe how f (x) =
behaves for x near 2, for the two cases x > 2 and x < 2.
x 2
The values get large and positive on one side, large and negative on the other, so for x coming from
the right, “the value approaches 1”, while from the left, “the value approaches 1”.
Combining the above ideas and notation of one sided limits and infinite limits, we state this as
lim
x!2
1
x
2
=
1,
lim
x!2+
1
x
2
= 1.
But the limits from the two sides are different, so
lim
x!2
1
x
2
Does Not Exist (DNE).
Examples 9,10
Study Exercises Do Exercises 1-4, 6⇤ , 15, 18⇤ , 19, 27, 28⇤ ; review all Examples.